Are there any theorems related to the product of Jacobi/Legendre Polynomials and/or Hypergeometric functions? Specifically, I'm interested in the product of ${}\_{2}F_{1}[-n,-n+1;2;x]$ and ${}\_{2}F_{1}[-n-1,-n+3;2;x]$ hoping to obtain it in some form ${}_{p}F_{q}$.

I've found some stuff in Bailey (1928,1935), but it has solutions only for some special cases. I've also obtained the coefficient of k'th term $\frac{x^k}{k!}$. I get (in case I didn't make any mistakes)

$\sum_{m=0}^{k} \binom{k}{m} \binom{n}{m}\binom{n-1}{m}\binom{n+1}{k-m}\binom{n-3}{k-m} \frac{m!(k-m)!}{(m+1)(k-m+1)}$, but I don't quite see what to do next.

2 Answers
2

Of course, there is no general formula of the type you wanted
but a whole bunch of the formulae expressing the product of
two $_2F_1$ by hypergeometric (or nearly hypergeometric) means.
They are known as Orr-type theorems and can be found in
Slater's book "Generalized hypergeometric functions", Section 2.5
(there are some instances in Bailey's "Generalized hypergeometric series"
as well). A famous example of this type is Clausen's identity
$$
{}_2F_1(a,b;a+b+\tfrac12\mid z)^2
={}_3F_2(2a,2b,a+b;2a+2b,a+b+\tfrac12\mid z).
$$
In addition, you can use the contiguous relations
[Slater, Section 1.4] which allow one to produce linear
relations between any three functions of the form
${}_2F_1(a+n,b+m;c+k\mid z)$ where $n,m,k\in\mathbb Z$,
as well as the transformation [Slater, Section 1.7.1]
$$
{}_2F_1(a,b;c\mid z)
=(1-z)^{-a}{}_2F_1\biggl(a,c-b;c\Bigm|\frac{z}{z-1}\biggr).
$$

I do not see however that something spectacular happens for
your particular product. In fact, the algorithm described
in the (already mentioned) book "$A=B$" decides whether the
expression $a_k$ given by
$$
\sum_{k=0}^\infty a_kz^k
:={}_2F_1(-n,-n+1;2\mid z){}_2F_1(-n-1,-n+3;2\mid z)
$$
(so that each $a_k$ is a single hypergeometric sum) can be
represented as a sum of finite rational terms. If this is
the case (which I really doubt), then you will have your wanted
product as a finite sum of hypergeometric functions; if not,
then this is the proof that you have no expression of this type.

Have you tried to look at the recurrence relation satisfied by the product of Jacobi polynomials?
(For example, maple/gfun finds product recurrences, but a list of related packages is on the site http://www.mat.univie.ac.at/~slc/divers/software.html of the Seminaire Lotharingien de Combinatoire.)

In general, this will not be a single sum, but a lot of things can be done just by using the recurrence relation instead of a sum representation and there is the Petkovsek algorithm to check if a nice closed-form solution exists.